Bring a large pot of lightly salted water to a boil. Cook bow-tie pasta at a boil, stirring occasionally, about 9 minutes. Add asparagus and cook until pasta is tender yet firm to the bite, about 3 minutes more; drain.
Heat olive oil in a large skillet over medium heat; cook and stir mushrooms, red bell pepper, and garlic until tender, 5 to 10 minutes. Reduce heat and stir in chicken, artichoke hearts, pesto, salt, and red pepper flakes; cook and stir until heated through, about 5 minutes.
Mix pasta and asparagus into chicken mixture, adding chicken broth if some moisture is needed. Sprinkle Parmesan cheese over the top.